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We analyze the thermalization properties and the validity of the Eigenstate Thermalization Hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a Z_2 symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the off-diagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of the observables on the energy eigenstates with respect to the dense ones. In particular, the numerical data seems to support the existence of rare states, i.e. states where the observables take expectation values which are different compared to the typical ones sampled by the micro-canonical distribution. In the case of sparse random matrices we also extract the finite size behavior of two different time scales associated with the thermalization process.
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We study quench dynamics and defect production in the Kitaev and the extended Kitaev models. For the Kitaev model in one dimension, we show that in the limit of slow quench rate, the defect density n sim 1/sqrt{tau} where 1/tau is the quench rate. We also compute the defect correlation function by providing an exact calculation of all independent non-zero spin correlation functions of the model. In two dimensions, where the quench dynamics takes the system across a critical line, we elaborate on the results of earlier work [K. Sengupta, D. Sen and S. Mondal, Phys. Rev. Lett. 100, 077204 (2008)] to discuss the unconventional scaling of the defect density with the quench rate. In this context, we outline a general proof that for a d dimensional quantum model, where the quench takes the system through a d-m dimensional gapless (critical) surface characterized by correlation length exponent u and dynamical critical exponent z, the defect density n sim 1/tau^{m u /(z u +1)}. We also discuss the variation of the shape and the spatial extent of the defect correlation function with the change of both the rate of quench and the model parameters and compute the entropy generated during such a quench process. Finally, we study the defect scaling law, entropy generation and defect correlation function of the two-dimensional extended Kitaev model.
We study the nonequilibrium dynamics of the extended toric code model (both ordered and disordered) to probe the existence of the dynamical quantum phase transitions (DQPTs). We show that in the case of the ordered toric code model, the zeros of Loschmidt overlap (generalized partition function) occur at critical times when DQPTs occur, which is confirmed by the nonanalyticities in the dynamical counter-part of the free-energy density. Moreover, we show that DQPTs occur for any non-zero field strength if the initial state is the excited state of the toric code model. In the disordered case, we show that it is imperative to study the behavior of the first time derivative of the dynamical free-energy density averaged over all the possible configurations, to characterize the occurrence of a DQPTs in the disordered toric code model since the disorder parameter itself acts as a new artificial dimension. We also show that for the case where anyonic excitations are present in the initial state, the conditions for a DQPTs to occur are the same as what happens in the absence of any excitation.
We consider quantum quenches in an integrable quantum chain with tuneable-integrability-breaking interactions. In the case where these interactions are weak, we demonstrate that at intermediate times after the quench local observables relax to a prethermalized regime, which can be described by a density matrix that can be viewed as a deformation of a generalized Gibbs ensemble. We present explicit expressions for the approximately conserved charges characterizing this ensemble. We do not find evidence for a crossover from the prethermalized to a thermalized regime on the time scales accessible to us. Increasing the integrability-breaking interactions leads to a behaviour that is compatible with eventual thermalization.
After a brief introduction to the concept of entanglement in quantum systems, I apply these ideas to many-body systems and show that the von Neumann entropy is an effective way of characterising the entanglement between the degrees of freedom in different regions of space. Close to a quantum phase transition it has universal features which serve as a diagnostic of such phenomena. In the second part I consider the unitary time evolution of such systems following a `quantum quench in which a parameter in the hamiltonian is suddenly changed, and argue that finite regions should effectively thermalise at late times, after interesting transient effects.
We study the dynamical response of a system to a sudden change of the tuning parameter $lambda$ starting (or ending) at the quantum critical point. In particular we analyze the scaling of the excitation probability, number of excited quasiparticles, heat and entropy with the quench amplitude and the system size. We extend the analysis to quenches with arbitrary power law dependence on time of the tuning parameter, showing a close connection between the scaling behavior of these quantities with the singularities of the adiabatic susceptibilities of order $m$ at the quantum critical point, where $m$ is related to the power of the quench. Precisely for sudden quenches the relevant susceptibility of the second order coincides with the fidelity susceptibility. We discuss the generalization of the scaling laws to the finite temperature quenches and show that the statistics of the low-energy excitations becomes important. We illustrate the relevance of those results for cold atoms experiments.
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